Non-negative Sparse Principal Component Analysis for Multidimensional Constrained Optimization

Abstract

One classic problem in air traffic management (ATM) has been the problem of detection and resolution of conflicts between aircraft. Traditionally, a conflict between two aircraft is detected whenever the two protective cylinders surrounding the aircraft intersect. In Trajectory-based Air Traffic Management, a baseline for the next generation of air traffic management system, we suggest that these protective cylinders be deformable volumes induced by variations in weather information such as wind speed and directions subjected to uncertainties of future states of trajectory controls. Using contact constraints on deforming parametric surfaces of these protective volumes, a constrained minimization algorithm is proposed to compute collision between two deformable bodies, and a differential optimization scheme is proposed to resolve detected conflicts. Given the covariance matrix representing the state of aircraft trajectory and its control and objective functions, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables where the coefficients in this combination are required to be non-negative, and the number of non-zero coefficients is constrained (e.g. state of trajectory and estimated time of arrival over one change point). Using convex relaxation and re-weighted l 1 technique, we reduce the problem to solving some semi-definite programming ones, and reinforce the non-negative principal components that satisfy the sparsity constraints. Numerical results show that the method presented in this paper is efficient and reliable in practice. Since the proposed method can be applied to a wide range of dynamic modeling problems such as collision avoidance in dynamic autonomous robots environments, dynamic interactions with 4D computer animation scenes, financial asset trading, or autonomous intelligent vehicles, we also attempt to keep all descriptions as general as possible.